{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2006:K5A56IRZEMQDUZDWYY6Q7UWGXR","short_pith_number":"pith:K5A56IRZ","schema_version":"1.0","canonical_sha256":"5741df223923203a6476c63d0fd2c6bc5630bc4079078817f2c9332d8d79649a","source":{"kind":"arxiv","id":"math/0612407","version":1},"attestation_state":"computed","paper":{"title":"Dynamics of the third order Lyness' difference equation","license":"","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Anna Cima, Armengol Gasull, Victor Manosa","submitted_at":"2006-12-14T17:46:23Z","abstract_excerpt":"This paper studies the iterates of the third order Lyness' recurrence $x_{k+3}=(a+x_{k+1}+x_{k+2})/x_k,$ with positive initial conditions, being $a$ also a positive parameter. It is known that for $a=1$ all the sequences generated by this recurrence are 8-periodic. We prove that for each $a\\ne1$ there are infinitely many initial conditions giving rise to periodic sequences which have almost all the even periods and that for a full measure set of initial conditions the sequences generated by the recurrence are dense in either one or two disjoint bounded intervals of $\\R.$ Finally we show that t"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"math/0612407","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.DS","submitted_at":"2006-12-14T17:46:23Z","cross_cats_sorted":[],"title_canon_sha256":"26eba06af90d7a7497b61e81abdec9cfeafa63501639b1782a09013367a853f2","abstract_canon_sha256":"01a9e303fa9d961834f4da14e22fbb772b98f3e1ac86a67053c12f82d962b23d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:32:43.977402Z","signature_b64":"IikuWePaSnIgtmlN3WNkOV80RyrpH8dcpGs/sGhqx6Dg8Be9qVDxCPWLpgx3CxpA8xC/mIcH1BuJiZV0tI/rAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5741df223923203a6476c63d0fd2c6bc5630bc4079078817f2c9332d8d79649a","last_reissued_at":"2026-05-18T04:32:43.976947Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:32:43.976947Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Dynamics of the third order Lyness' difference equation","license":"","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Anna Cima, Armengol Gasull, Victor Manosa","submitted_at":"2006-12-14T17:46:23Z","abstract_excerpt":"This paper studies the iterates of the third order Lyness' recurrence $x_{k+3}=(a+x_{k+1}+x_{k+2})/x_k,$ with positive initial conditions, being $a$ also a positive parameter. It is known that for $a=1$ all the sequences generated by this recurrence are 8-periodic. We prove that for each $a\\ne1$ there are infinitely many initial conditions giving rise to periodic sequences which have almost all the even periods and that for a full measure set of initial conditions the sequences generated by the recurrence are dense in either one or two disjoint bounded intervals of $\\R.$ Finally we show that t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0612407","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"math/0612407","created_at":"2026-05-18T04:32:43.977012+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0612407v1","created_at":"2026-05-18T04:32:43.977012+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0612407","created_at":"2026-05-18T04:32:43.977012+00:00"},{"alias_kind":"pith_short_12","alias_value":"K5A56IRZEMQD","created_at":"2026-05-18T12:25:54.717736+00:00"},{"alias_kind":"pith_short_16","alias_value":"K5A56IRZEMQDUZDW","created_at":"2026-05-18T12:25:54.717736+00:00"},{"alias_kind":"pith_short_8","alias_value":"K5A56IRZ","created_at":"2026-05-18T12:25:54.717736+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/K5A56IRZEMQDUZDWYY6Q7UWGXR","json":"https://pith.science/pith/K5A56IRZEMQDUZDWYY6Q7UWGXR.json","graph_json":"https://pith.science/api/pith-number/K5A56IRZEMQDUZDWYY6Q7UWGXR/graph.json","events_json":"https://pith.science/api/pith-number/K5A56IRZEMQDUZDWYY6Q7UWGXR/events.json","paper":"https://pith.science/paper/K5A56IRZ"},"agent_actions":{"view_html":"https://pith.science/pith/K5A56IRZEMQDUZDWYY6Q7UWGXR","download_json":"https://pith.science/pith/K5A56IRZEMQDUZDWYY6Q7UWGXR.json","view_paper":"https://pith.science/paper/K5A56IRZ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0612407&json=true","fetch_graph":"https://pith.science/api/pith-number/K5A56IRZEMQDUZDWYY6Q7UWGXR/graph.json","fetch_events":"https://pith.science/api/pith-number/K5A56IRZEMQDUZDWYY6Q7UWGXR/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/K5A56IRZEMQDUZDWYY6Q7UWGXR/action/timestamp_anchor","attest_storage":"https://pith.science/pith/K5A56IRZEMQDUZDWYY6Q7UWGXR/action/storage_attestation","attest_author":"https://pith.science/pith/K5A56IRZEMQDUZDWYY6Q7UWGXR/action/author_attestation","sign_citation":"https://pith.science/pith/K5A56IRZEMQDUZDWYY6Q7UWGXR/action/citation_signature","submit_replication":"https://pith.science/pith/K5A56IRZEMQDUZDWYY6Q7UWGXR/action/replication_record"}},"created_at":"2026-05-18T04:32:43.977012+00:00","updated_at":"2026-05-18T04:32:43.977012+00:00"}